#72342793

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting

C₂×#21483 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₈	C₂×#1020⋊C₂
commutator, hypocenter	#997	C₂³×C₄
Frattini	C₂²×C₄	C₂³×S₄
Fitting	C₂×#21483	D₃
socle	C₂	C₂×C₄×#1020⋊C₂
prime radical	#72236838	C₂
cosocle	#5950	C₂⁴
hypercenter	C₂×C₈	#1020⋊C₂
LCS	G ▷ #997
UCS	C₁ ◁ C₈ ◁ C₂×C₈

5948 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,22,24,20,7,6,21,5)(2,34,36,32,12,11,33,10)(3,16,23,15,13,39,42,38)(4,45,47,43,19,18,44,17)(8,28,35,27,25,50,53,49)(9,56,58,54,31,30,55,29)(14,41,46,40,37,60,61,59)(26,52,57,51,48,63,64,62);(1,3)(2,8)(4,14)(5,15)(6,23)(7,13)(9,26)(10,27)(11,35)(12,25)(16,24)(17,40)(18,46)(19,37)(20,38)(21,39)(22,42)(28,36)(29,51)(30,57)(31,48)(32,49)(33,50)(34,53)(41,47)(43,59)(44,60)(45,61)(52,58)(54,62)(55,63)(56,64);(1,2,4,9,7,12,19,31)(3,8,14,26,13,25,37,48)(5,10,17,29,20,32,43,54)(6,11,18,30,22,34,45,56)(15,27,40,51,38,49,59,62)(16,28,41,52,39,50,60,63)(21,33,44,55,24,36,47,58)(23,35,46,57,42,53,61,64);(5,20)(6,22)(10,32)(11,34)(15,38)(16,39)(17,43)(18,45)(21,24)(23,42)(27,49)(28,50)(29,54)(30,56)(33,36)(35,53)(40,59)(41,60)(44,47)(46,61)(51,62)(52,63)(55,58)(57,64)

Polycyclic presentation:

2^5,3,2^5|b^2=e,e^2=k,g^2=k,h^2=k,d^c=dk,f^c=f^2,f^d=f^2,g^a=gjk,g^c=gj,g^d=gk,g^f=h,h^a=hik,h^c=ghijk,h^d=gh,h^f=ghk,h^g=hk,i^c=ij,i^d=ij,i^f=ij,i^g=ik,j^f=i,j^h=jk

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342856
maximal normal subgroup of	#72342856

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 5, F.11);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 6, 4, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.11);;
gap> SetConjugate(coll, 8, 3, F.7*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 6, F.7*F.8*F.11);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 10, 6, F.9);;
gap> SetConjugate(coll, 10, 8, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);